math::changepoint  Change point detection methods
package require
Tcl 8.6
package require
TclOO
package require
math::statistics
package require
math::changepoint ?0.1?
::math::changepoint::cusumdetect data ?args?
::math::changepoint::cusumonline ?args?
$cusumObj examine
value
$cusumObj reset
::math::changepoint::binarysegmentation data ?args?
The
math::changepoint package implements a number of wellknown methods
to determine if a series of data contains a shift in the mean or not. Note
that these methods only indicate if a shift in the mean is probably. Due to
the stochastic nature of the data that will be analysed, false positives are
possible. The CUSUM method is implemented in both an "offline" and
an "online" version, so that it can be used either for a complete
data series or for detecting changes in data that come in one by one. The
implementation has been based on these websites mostly:
 •
 https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc323.htm
 •
 https://en.wikipedia.org/wiki/CUSUM
Basically, the deviation of the data from a given target value is accumulated
and when the total deviation becomes too large, a change point is reported. A
second method, binary segmentation, is implemented only as an
"offline" method, as it needs to examine the data series as a whole.
In the variant contained here the following ideas have been used:
 •
 The segments in which the data series may be separated
shold not be too short, otherwise the ultimate result could be segments of
only one data point long. So a minimum length is used.
 •
 To make the segmentation worthwhile there should be a
minimum gain in reducing the cost function (the sum of the squared
deviations from the mean for each segment).
This may not be in agreement with the descriptions of the method found in
various publications, but it is simple to understand and intuitive. One
publication that provides more information on the method in general is
"Selective review of offline change point detection methods" by
Truong et al.
https://arxiv.org/abs/1801.00718.
The package defines the following public procedures:

::math::changepoint::cusumdetect data
?args?
 Examine a given data series and return the location of the
first change (if any)
 double data
 Series of data to be examined
 list args
 Optional list of keyvalue pairs:

target value
 The target (or mean) for the time series

tolerance value
 The tolerated standard deviation

kfactor value
 The factor by which to multiply the standard deviation
(defaults to 0.5, typically between 0.5 and 1.0)

hfactor value
 The factor determining the limits betweem which the
"cusum" statistic is accepted (typicaly 3.05.0, default
4.0)

::math::changepoint::cusumonline ?args?
 Class to examine data passed in against expected
properties. At least the keywords target and tolerance
must be given.
 list args
 List of keyvalue pairs:

target value
 The target (or mean) for the time series

tolerance value
 The tolerated standard deviation

kfactor value
 The factor by which to multiply the standard deviation
(defaults to 0.5, typically between 0.5 and 1.0)

hfactor value
 The factor determining the limits betweem which the
"cusum" statistic is accepted (typicaly 3.05.0, default
4.0)

$cusumObj examine value
 Pass a value to the cusumonline object and examine
it. If, with this new value, the cumulative sum remains within the bounds,
zero (0) is returned, otherwise one (1) is returned.
 double value
 The new value

$cusumObj reset
 Reset the cumulative sum, so that the examination can start
afresh.

::math::changepoint::binarysegmentation data
?args?
 Apply the binary segmentation method recursively to find
change points. Returns a list of indices of potential change points
 list data
 Data to be examined
 list args
 Optional keyvalue pairs:

minlength number
 Minimum number of points in each segment (default: 5)

threshold value
 Factor applied to the standard deviation functioning as a
threshold for accepting the change in cost function as an improvement
(default: 1.0)
control, statistics
Mathematics
Copyright (c) 2020 by Arjen Markus